Simplifying numerical solution of constrained PDE systems through involutive completion

Bijan Mohammadi; Jukka Tuomela

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2005)

  • Volume: 39, Issue: 5, page 909-929
  • ISSN: 0764-583X

Abstract

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When analysing general systems of PDEs, it is important first to find the involutive form of the initial system. This is because the properties of the system cannot in general be determined if the system is not involutive. We show that the notion of involutivity is also interesting from the numerical point of view. The use of the involutive form of the system allows one to consider quite general situations in a unified way. We illustrate our approach on the numerical solution of several flow equations with the aim of showing the impact of the involutive form of the systems in simplifying numerical schemes.

How to cite

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Mohammadi, Bijan, and Tuomela, Jukka. "Simplifying numerical solution of constrained PDE systems through involutive completion." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 39.5 (2005): 909-929. <http://eudml.org/doc/244671>.

@article{Mohammadi2005,
abstract = {When analysing general systems of PDEs, it is important first to find the involutive form of the initial system. This is because the properties of the system cannot in general be determined if the system is not involutive. We show that the notion of involutivity is also interesting from the numerical point of view. The use of the involutive form of the system allows one to consider quite general situations in a unified way. We illustrate our approach on the numerical solution of several flow equations with the aim of showing the impact of the involutive form of the systems in simplifying numerical schemes.},
author = {Mohammadi, Bijan, Tuomela, Jukka},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {overdetermined PDEs; involution; discretization; elliptic systems; involutive form; overdetermined systems; auxiliary variables},
language = {eng},
number = {5},
pages = {909-929},
publisher = {EDP-Sciences},
title = {Simplifying numerical solution of constrained PDE systems through involutive completion},
url = {http://eudml.org/doc/244671},
volume = {39},
year = {2005},
}

TY - JOUR
AU - Mohammadi, Bijan
AU - Tuomela, Jukka
TI - Simplifying numerical solution of constrained PDE systems through involutive completion
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2005
PB - EDP-Sciences
VL - 39
IS - 5
SP - 909
EP - 929
AB - When analysing general systems of PDEs, it is important first to find the involutive form of the initial system. This is because the properties of the system cannot in general be determined if the system is not involutive. We show that the notion of involutivity is also interesting from the numerical point of view. The use of the involutive form of the system allows one to consider quite general situations in a unified way. We illustrate our approach on the numerical solution of several flow equations with the aim of showing the impact of the involutive form of the systems in simplifying numerical schemes.
LA - eng
KW - overdetermined PDEs; involution; discretization; elliptic systems; involutive form; overdetermined systems; auxiliary variables
UR - http://eudml.org/doc/244671
ER -

References

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  1. [1] M.S. Agranovich, Elliptic boundary problems, Partial differential equations IX. M.S. Agranovich, Yu.V. Egorov and M.A. Shubin, Eds., Springer. Encyclopaedia Math. Sci. 79 (1997) 1–144. Zbl0880.35001
  2. [2] Å. Björck, Numerical methods for least squares problems, SIAM (1996). Zbl0847.65023MR1386889
  3. [3] H. Borouchaki, P.L. George and B. Mohammadi, Delaunay mesh generation governed by metric specifications. Parts i & ii. Finite Elem. Anal. Des., Special Issue on Mesh Adaptation (1996) 345–420. 
  4. [4] M. Castro-Diaz, F. Hecht and B. Mohammadi, Anisotropic grid adaptation for inviscid and viscous flows simulations. Int. J. Numer. Meth. Fl. 25 (1995) 475–491. Zbl0902.76057
  5. [5] A. Douglis and L. Nirenberg, Interior estimates for elliptic systems of partial differential equations. Comm. Pure Appl. Math. 8 (1955) 503–538. Zbl0066.08002
  6. [6] P.I. Dudnikov and S.N. Samborski, Linear overdetermined systems of partial differential equations. Initial and initial-boundary value problems, Partial Differential Equations VIII, M.A. Shubin, Ed., Springer-Verlag, Berlin/Heidelberg. Encyclopaedia Math. Sci. 65 (1996) 1–86. Zbl0831.35113
  7. [7] Femlab 3.0, http://www.comsol.com/products/femlab/ 
  8. [8] FreeFem, http://www.freefem.org/ 
  9. [9] P.L. George, Automatic mesh generation. Applications to finite element method, Wiley (1991). Zbl0808.65122MR1165297
  10. [10] R. Glowinski, Finite element methods for incompressible viscous flow. Handb. Numer. Anal. Vol. IX, North-Holland, Amsterdam (2003) 3–1176. Zbl1040.76001
  11. [11] F. Hecht and B. Mohammadi, Mesh adaptation by metric control for multi-scale phenomena and turbulence. American Institute of Aeronautics and Astronautics 97-0859 (1997). 
  12. [12] B. Jiang, J. Wu and L. Povinelli, The origin of spurious solutions in computational electromagnetics. J. Comput. Phys. 7 (1996) 104–123. Zbl0848.65086
  13. [13] K. Krupchyk, W. Seiler and J. Tuomela, Overdetermined elliptic PDEs. J. Found. Comp. Math., submitted. Zbl1101.35061
  14. [14] E.L. Mansfield, A simple criterion for involutivity. J. London Math. Soc. (2) 54 (1996) 323–345. Zbl0865.35092
  15. [15] B. Mohammadi and J. Tuomela, Involutivity and numerical solution of PDE systems, in Proc. of ECCOMAS 2004, Vol. 1, Jyväskylä, Finland. P. Neittaanmäki, T. Rossi, K. Majava and O. Pironneau, Eds., University of Jyväskylä (2004) 1–10. 
  16. [16] F. Nicoud, Conservative high-order finite-difference schemes for low-Mach number flows. J. Comput. Phys. 158 (2000) 71–97. Zbl0973.76068
  17. [17] O. Pironneau, Finite element methods for fluids, Wiley (1989). Zbl0712.76001MR1030279
  18. [18] J.F. Pommaret, Systems of partial differential equations and Lie pseudogroups. Math. Appl., Gordon and Breach Science Publishers 14 (1978). Zbl0401.58006MR517402
  19. [19] R.F. Probstein, Physicochemical hydrodynamics, Wiley (1995). 
  20. [20] A. Quarteroni and A. Valli, Numerical approximation of partial differential equations. Springer Ser. Comput. Math. 23 (1994). Zbl0803.65088MR1299729
  21. [21] W.M. Seiler, Involution — the formal theory of differential equations and its applications in computer algebra and numerical analysis, Habilitation thesis, Dept. of Mathematics, Universität Mannheim (2001) (manuscript accepted for publication by Springer-Verlag). 
  22. [22] D. Spencer, Overdetermined systems of linear partial differential equations. Bull. Am. Math. Soc. 75 (1969) 179–239. Zbl0185.33801
  23. [23] J. Tuomela and T. Arponen, On the numerical solution of involutive ordinary differential systems. IMA J. Numer. Anal. 20 (2000) 561–599. Zbl0982.65088
  24. [24] J. Tuomela and T. Arponen, On the numerical solution of involutive ordinary differential systems: Higher order methods. BIT 41 (2001) 599–628. Zbl1001.65093
  25. [25] J. Tuomela, T. Arponen and V. Normi, On the numerical solution of involutive ordinary differential systems: Enhanced linear algebra. IMA J. Numer. Anal., submitted. Zbl1105.65083

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